To make that work, you would have to change the way your site works - it couldn’t just be static HTML. You would have to add a timer in the client-side JS that checks for changes and causes the page to reload if changes are detected.

When you click “Show Live”, the Glitch editor creates the other tab that displays the website, and it still has a handle to that window. Then, when you make changes, the editor can tell that tab to reload using the handle. If you create a new tab on your own and point it at your application in the same browser, the editor won’t know about it, and won’t reload it when you make changes.